Tusnády’s inequality revisited
Andrew Carter and David Pollard
Source: Ann. Statist. Volume 32, Number 6
(2004), 2731-2741.
Abstract
Tusnády’s inequality is the key ingredient in the KMT/Hungarian coupling of the empirical distribution function with a Brownian bridge. We present an elementary proof of a result that sharpens the Tusnády inequality, modulo constants. Our method uses the beta integral representation of Binomial tails, simple Taylor expansion and some novel bounds for the ratios of normal tail probabilities.
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Keywords: Quantile coupling; KMT/Hungarian construction; Tusnády’s inequality; beta integral representation of Binomial tails; ratios of normal tails; equivalent normal deviate
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1107794885
Digital Object Identifier: doi:10.1214/009053604000000733
Mathematical Reviews number (MathSciNet): MR2154001
Zentralblatt MATH identifier: 1076.62012
References
Bretagnolle, J. and Massart, P. (1989). Hungarian constructions from the nonasymptotic viewpoint. Ann. Probab. 17 239--256.
Mathematical Reviews (MathSciNet): MR972783
JSTOR: links.jstor.org
Brown, L. D., Carter, A. V., Low, M. G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 2074--2097.
Mathematical Reviews (MathSciNet): MR2102503
Digital Object Identifier: doi:10.1214/009053604000000454
Project Euclid: euclid.aos/1098883782
Zentralblatt MATH: 1062.62083
Csörgő, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York.
Mathematical Reviews (MathSciNet): MR666546
de Bruijn, N. G. (1981). Asymptotic Methods in Analysis, corrected 3rd ed. Dover, New York.
Mathematical Reviews (MathSciNet): MR671583
Zentralblatt MATH: 0556.41021
Dudley, R. M. (2000). Notes on empirical processes. Lecture notes for a course given at Aarhus Univ., August 1999.
Feller, W. (1968). An Introduction to Probability Theory and Its Applications 1, 3rd ed. Wiley, New York.
Mathematical Reviews (MathSciNet): MR228020
Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent rv's and the sample df. I. Z. Wahrsch. Verw. Gebiete 32 111--131.
Mathematical Reviews (MathSciNet): MR375412
Digital Object Identifier: doi:10.1007/BF00533093
Major, P. (2000). The approximation of the normalized empirical ditribution function by a Brownian bridge. Technical report, Mathematical Institute of the Hungarian Academy of Sciences. Notes available from www.renyi.hu/~major/.
Mason, D. M. (2001). Notes on the KMT Brownian bridge approximation to the uniform empirical process. In Asymptotic Methods in Probability and Statistics with Applications (N. Balakrishnan, I. A. Ibragimov and V. B. Nevzorov, eds.) 351--369. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet): MR1890338
Massart, P. (2002). Tusnády's lemma, 24 years later. Ann. Inst. H. Poincaré Probab. Statist. 38 991--1007.
Mathematical Reviews (MathSciNet): MR1955348
Digital Object Identifier: doi:10.1016/S0246-0203(02)01130-5
Molenaar, W. (1970). Approximations to the Poisson, Binomial, and Hypergeometric Distribution Functions. Math. Centrum, Amsterdam.
Mathematical Reviews (MathSciNet): MR275635
Zentralblatt MATH: 0199.53001
Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399--2430.
Mathematical Reviews (MathSciNet): MR1425959
Digital Object Identifier: doi:10.1214/aos/1032181160
Project Euclid: euclid.aos/1032181160
Peizer, D. B. and Pratt, J. W. (1968). A normal approximation for Binomial, F, beta, and other common, related tail probabilities. I. J. Amer. Statist. Assoc. 63 1416--1456.
Mathematical Reviews (MathSciNet): MR235650
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
Mathematical Reviews (MathSciNet): MR388499
Pratt, J. W. (1968). A normal approximation for Binomial, F, beta, and other common, related tail probabilities. II. J. Amer. Statist. Assoc. 63 1457--1483.
Mathematical Reviews (MathSciNet): MR235651
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
Mathematical Reviews (MathSciNet): MR838963
Tusnády, G. (1977). A study of statistical hypotheses. Ph. D. thesis, Hungarian Academy of Sciences, Budapest.
